Basic invariants
| Dimension: | $1$ |
| Group: | $C_2$ |
| Conductor: | \(1077\)\(\medspace = 3 \cdot 359 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of \(\Q(\sqrt{1077}) \) |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_2$ |
| Parity: | even |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 + 13\cdot 29 + 18\cdot 29^{2} + 28\cdot 29^{3} + 11\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 + 15\cdot 29 + 10\cdot 29^{2} + 17\cdot 29^{4} +O(29^{5})\)
|
Generators of the action on the roots $ r_{ 1 }, r_{ 2 } $
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $ r_{ 1 }, r_{ 2 } $ | Character values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)$ | $-1$ |